Single-cell and extracellular nano-vesicles biosensing through phase spectral analysis of optical fiber tweezers back-scattering signals

Diagnosis of health disorders relies heavily on detecting biological data and accurately observing pathological changes. A significant challenge lies in detecting targeted biological signals and developing reliable sensing technology for clinically relevant results. The combination of data analytics with the sensing abilities of Optical Fiber Tweezers (OFT) provides a high-capability, multifunctional biosensing approach for biophotonic tools. In this work, we introduced phase as a new domain to obtain light patterns in OFT back-scattering signals. By applying a multivariate data analysis procedure, we extract phase spectral information for discriminating micro and nano (bio)particles. A newly proposed method—Hilbert Phase Slope—presented high suitability for differentiation problems, providing features able to discriminate with statistical significance two optically trapped human tumoral cells (MKN45 gastric cell line) and two classes of non-trapped cancer-derived extracellular nanovesicles – an important outcome in view of the current challenges of label-free bio-detection for multifunctional single-molecule analytic tools.

To verify the influence of time shifts in the phase, an approach presented in [1] was followed, based on the analysis of a signal with left-right symmetry.When a signal presents this symmetric property with respect to the origin, the phase is linear and equal to zero.By shifting the signal in time, a change in the slope of the linear phase must be observed.To test this, two variations of the first signal were created, as illustrated in Supplementary Figure 2 (A), were a 0.5 second time delay and advance were introduced.This results in the original signal shifted to the left and to the right, respectively [1].The spectral representations obtained after applying FFT, small magnitude values removal, phase calculation and unwrapping are presented on Supplementary Figure 2 (B).After phase calculation, the discontinuities present were compensated through the phase unwrapping algorithm.As it was expected, a linear phase response is obtained from the symmetrical time domain waveform.When the peak is centered on sample N/2, it presents zero phase.By shifting in the time domain, a changing in the slope of the linear phase is observed [1].Therefore, it confirms that the algorithm applied provides a correct phase representation and can then be used to extract phase spectrum from the back-scattered signals.
A new method is presented to obtain phase spectrum -Absolute Hilbert Phase Slope -where phase is calculated at each sampling instant relative to the phase position of the previous point in time, along the horizontal direction (x-direction).To evaluate the suitability of this approach to study phase patterns, a validation experiment was conducted.A synthetic signal was used to calculate the phase of a Hilbert transformed signal (Supplementary Figure 3 (A)).The discontinuities were removed using the previously applied unwrapping algorithm, providing a continuous phase representation, as observed in a b Supplementary Figure 2 Illustration of the phase changes observed with time domain shifts.a) A time domain waveform symmetrical around a vertical axis was selected for this experiment since signals with this type of symmetry present a linear phase spectrum.After phase calculation and unwrapping, a linear phase is obtained as expected for the symmetrical signals used, which confirms that the algorithm applied provides a correct phase representation.b) When the time domain waveform is shifted to the right, the phase remains a straight line, but experiences a decrease in slope.When the time domain is shifted to the left, there is an increase in the slope.Such outcome is equally described for this experiment.This result confirms that the algorithm applied provides a correct phase representation and can then be used to extract phase spectrum from the backscattered signals.Besides, it illustrates how time-domain shifts influence phase spectrum, thus supporting the hypothesis that phase spectrum can retain patterns related with the bounce-back reflections contained in the back-scattering signal received and analyzed in this experiment.
Supplementary Figure 3 (B).An interesting observation was registered when calculating the absolute value of unwrapped phase slope.Larger phase shifts were observed in comparison with the wrapped phase wraps observed, that seem to be masked in this calculation (Supplementary Figure 3 (C)).By calculating the phase slope between adjacent points, larger phase shifts were observed, which may be correlated with larger reflections in our experiment.This way, this method seems to provide a resulting spectrum more sensitive and robust to discriminate phase oscillations.

Supplementary Note 2: Mathematical Formulations
Detailed explanation of signal processing procedure with underlying mathematical formulations.

Eq. (1)
The Discrete Fourier Transform is calculated by where () is the signal value at sample n and $% is a sequence of  complex coefficients [2].This operation was conducted through the efficient FFT algorithm.For each complex coefficients, the phase parameter was calculated from the inverse tangent function: ∅() = arctan : ()(+(,)) ./ (+(,)) ; Eq. ( 3) with  and  being the imaginary and real components of the complex Fourier spectrum of the data.This calculation retrieves a phase distribution forced to the principal value range, − < ∅() <  Eq. ( 4) Requiring a phase unwrapping algorithm to determine discontinuities on the wrapped phase, resolving them, and achieving a continuous phase spectrum [3].Mathematically, this phase unwrapping operation is described as [4]: Eq. ( 5) where () is the estimated unwrapped phase, ∅() the wrapped phase obtained from Equation ( 2), and () is the integer value that specifies the corrective offset required.The procedure is based on transversing, in the x direction, through the wrapped phase vector to detect the presence of discontinuities between adjacent samples, that are compensated with a 2π addition or 2π subtraction when a difference larger than +π or smaller than -π, respectively, is found.This procedure allows removing the discontinuities to obtain a continuous phase signal [4].
The second method used to analyze phase was based on Hilbert Transform (), related to the Fourier Transform  for the discrete-time signal  as [5], [6] (())() =  1 () + ()() Eq. ( 6) where Eq. (7) meaning that negative frequencies are multiplied by  while positive frequencies are multiplied by −, which will produce a 90º rotation in the complex plane, creating a phase-shifted version of the original signal [2].Hilbert Transform operation allows to calculate instantaneous attributes from the time series (), through the computation of the analytical signal [2], [6]: Eq. ( 8) where  !represents the real part and  " the imaginary part.This expression in polar form is equivalent to [2], [5] () = () "4 (5)   Eq. ( 9) where A(t) represents the instantaneous amplitude and () the instantaneous phase, calculated by [2]: () = arctan : ; Eq. ( 10) Following the unwrapping procedure presented in Equation ( 5), the unwrapped instantaneous phase was calculated, and the two types of Hilbert-phase representations explored in this study were obtained: Instantaneous Phase and Instantaneous Phase Slope, calculated respectively by: Eq. ( 12)

Interquartile Range
Dispersion and dynamic range of the spectrum

Supplementary Figure 3
Validation experiment for Absolute Hilbert Phase Slope method, where phase is calculated at each sampling instant relative to the phase position of the previous point in time, along the horizontal direction (x-direction).The major phase shifts observed are numerated, for each phase representation: a) Hilbert wrapped phase, b) Hilbert Unwrapped phase and c) Absolute Hilbert Phase Slope.Although phase shifts 4 and 5 seem to present higher amplitude in the first two spectral representations, the absolute unwrapped phase slope revealed new patterns in the phase, where larger phase shifts are identified (e.g., shift 3), that may be hidden in the previous calculation.This result indicates that the algorithm allows a more precise extraction of the phase bounce-back reflection patterns between adjacent points.Therefore, is possible that through this method, light signatures with more detail and specificity are being obtained, creating a new phase representation with considerable improved content in discriminative patterns.